High-order iterative methods for a nonlinear Kirchhoff wave equation

• Autorzy: Ngoc L. T. P. , Truong L. X. , Long N. T.
• Języki publikacji: EN

Abstrakty

EN

In this paper we consider the following nonlinear wave equation (1) (…) where μ, f, ũ0, ũ1 are given functions satisfying conditions specified later. In Eq. (1)1, the nonlinear term μ(…) depends on the integral (…) dx. In this paper we associate with equation (1)1 a recurrent sequence {um} defined by (2) (…), 0< x <1, 0< t < T, with um satisfying (1)2,3. The first term u0 is chosen as u0≡0. If f∈CN([0,1]×R+×R), we prove that the sequence {um} converges at a rate of order N to a unique weak solution of problem (1).

Słowa kluczowe

EN

nonlinear Kirchhoff-Carrier wave equation; Faedo-Galerkin method; convergence of order N

PL

nieliniowe równania falowe; metoda Faedo-Galerkina

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2010
• Tom: Vol. 43, nr 3
• Strony: 605—634
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Ngoc L. T. P.

• Nhatrang Educational College 01 Nguyen Chanh Str. Nhatrang City, Vietnam

autor

Truong L. X.

• Department of Mathematics University of Economics Ho Chi Minh City 179-181, 3-2 Street Dist. 10, Ho Chi Minh City, Vietnam

autor

Long N. T.

• Department of Mathematics and Computer Science University of Natural Science Vietnam National University Ho Chi Minh City 227 Nguyen Van Cu Str. Dist.5, Ho Chi Minh City, Vietnam

Bibliografia

• [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
• [2] M. Aassila, Asymptotic behavior of solutions to a quasilinear hyperbolic equation with nonlinear damping, E. J. Qualitative Theory of Diff. Equ. 7 (1998), 1–12.
• [3] D. T. T. Binh, A. P. N. Dinh, N. T. Long, Linear recursive schemes associated with the nonlinear wave equation involving Bessel’s operator, Math. Comput. Modelling 34 (5–6) (2001), 541–556.
• [4] G. F. Carrier, On the nonlinear vibrations problem of elastic string, Quart. Appl. Math. 3 (1945), 157–165.
• [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, J. A. Soriano, Existence and exponential decay for a Kirchhoff–Carrier model with viscosity, J. Math. Anal. Appl. 226(1) (1998), 40–60.
• [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation, Adv. Differential Equations 6(6) (2001), 701–730.
• [7] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math. 6 (5) (2004), 705–731.
• [8] A. P. N. Dinh, N. T. Long, Linear approximation and asymptotic expansion associated to the nonlinear wave equation in one dimension, Demonstratio Math. 19 (1986), 45–63.
• [9] Y. Ebihara, L. A. Medeiros, M. M. Minranda, Local solutions for a nonlinear degenerate hyperbolic equation, Nonlinear Anal. 10 (1986), 27–40.
• [10] M. Hosoya, Y. Yamada, On some nonlinear wave equation I: Local existence and regularity of solutions, J. Fac. Sci. Univ. Tokyo. Sect. IA Math. 38 (1991), 225–238.
• [11] G. R. Kirchhoff, Vorlesungen über Mathematiche Physik: Mechanik, Teuber, Leipzig, 1876, Section 29.7.
• [12] V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, Vol. 1, Academic Press, New York, 1969.
• [13] I. Lasiecka, J. Ong, Global solvability and uniform decays of solutions to quasi linear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations 24 (11–12) (1999), 2069–2108.
• [14] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires, Dunod-Gauthier-Villars, Paris, 1969.
• [15] J. L. Lions, On some questions in boundary value problems of mathematical physics, in: G. de la Penha, L. A. Medeiros (Eds.), International Symposium on Continuum, Mechanics and Partial Differential Equations, Rio de Janeiro 1977, Mathematics Studies, vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346.
• [16] N. T. Long, T. N. Diem, On the nonlinear wave equation utt – uxx = f (x, t, u, ux, ut) associated with the mixed homogeneous conditions, Nonlinear Anal. 29(11) (1997), 1217–1230.
• [17] N. T. Long, A. P. N. Dinh, T. N. Diem, Linear recursive schemes and asymptotic expansion associated with the Kirchhoff-Carrier operator, J. Math. Anal. Appl. 267 (1) (2002), 116–134.
• [18] N. T. Long, On the nonlinear wave equation utt – B (t, ||ux||2) uxx=f(x, t, u, ux, ut) associated with the mixed homogeneous conditions, J. Math. Anal. Appl. 274 (1) (2002), 102–123.
• [19] N. T. Long, Nonlinear Kirchhoff–Carrier wave equation in a unit membrane with mixed homogeneous boundary conditions, Electron. J. Differential Equations 138 (2005), 18 pp.
• [20] N. T. Long, On the nonlinear wave equation utt−B(…) associated with the mixed homogeneous conditions, J. Math. Anal. Appl. 306 (1) (2005), 243–268.
• [21] N. T. Long, L. T. P. Ngoc, On a nonlinear Kirchhoff–Carrier wave equation in the unit membrane: The quadratic convergence and asymptotic expansion of solutions, Demonstratio Math. 40 (2007), 365–392.
• [22] L. A. Medeiros, On some nonlinear perturbation of Kirchhoff–Carrier operator, Comp. Appl. Math. 13 (1994), 225–233.
• [23] L. A. Medeiros, J. Limaco, S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, Part one, J. Comput. Anal. Appl. 4 (2) (2002), 91–127.
• [24] L. A. Medeiros, J. Limaco, S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, Part two, J. Comput. Anal. Appl. 4 (3) (2002), 211–263.
• [25] M. Milla Miranda, L. P. San Gil Jutuca, Existence and boundary stabilization of solutions for the Kirchhoff equation, Comm. Partial Differential Equations 24 (9-10) (1999), 1759–1800.
• [26] S. I. Pohozaev, On a class of quasilinear hyperbolic equation, Math. USSR. Sb. 25 (1975), 145–158.
• [27] E. L. Ortiz, A. P. N. Dinh, Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the Tau method, SIAM J. Math. Anal. 18 (1987), 452–464.